A300383 -id:A300383 - cp's OEIS frontend

A302094 Number of relatively prime or monic twice-partitions of n.

1, 3, 6, 10, 27, 35, 113, 170, 396, 641, 1649, 2318, 5905, 9112, 18678, 32529, 69094, 106210, 227480, 363433, 705210, 1196190, 2325023, 3724233, 7192245, 11915884, 21857887, 36597843, 67406158, 109594872, 201747847, 333400746, 591125465, 987069077, 1743223350

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime). Then a relatively prime or monic twice-partition of n is a choice of a relatively prime or monic partition of each part in a relatively prime or monic partition of n.

			The a(4) = 10 relatively prime or monic twice-partitions:
(4), (31), (211), (1111),
(3)(1), (21)(1), (111)(1),
(2)(1)(1), (11)(1)(1),
(1)(1)(1)(1).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A300486, A301462, A301467, A301480, A302915, A302916, A302917.

ip[n_]:=ip[n]=Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&];
Table[Sum[Times@@Length/@ip/@ptn,{ptn,ip[n]}],{n,10}]

A316091 Heinz numbers of integer partitions of prime numbers.

Original entry on oeis.org

3, 4, 5, 6, 8, 11, 14, 15, 17, 18, 20, 24, 26, 31, 32, 33, 35, 41, 42, 44, 45, 50, 54, 56, 58, 59, 60, 67, 69, 72, 74, 80, 83, 92, 93, 95, 96, 106, 109, 114, 119, 122, 124, 127, 128, 141, 143, 145, 152, 153, 157, 158, 161, 170, 174, 177, 179, 182, 188, 191

Offset: 1

Gus Wiseman, Jun 24 2018

nonn

Also the union of prime-indexed rows of A215366.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Sequence of all integer partitions of prime numbers begins (2), (1, 1), (3), (2, 1), (1, 1, 1), (5), (4, 1), (3, 2), (7), (2, 2, 1), (3, 1, 1), (2, 1, 1, 1), (6, 1).

Cf. A000041, A000607, A056239, A056768, A076610, A100118, A112798, A215366, A296150, A300383, A316092.

primeMS[n_] := If[n == 1, {}, Flatten[Cases[FactorInteger[n],{p_, k_} :> Table[PrimePi[p], {k}]]]]; Select[Range[100], PrimeQ[Total[primeMS[#]]] &]

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 9, 5, 9, 2, 23, 2, 11, 10, 24, 2, 33, 2, 36, 12, 15, 2, 87, 7, 17, 17, 53, 2, 96, 2, 79, 16, 21, 14, 196, 2, 23, 18, 154, 2, 166, 2, 99, 54, 27, 2, 431, 9, 85, 22, 128, 2, 303, 18, 261, 24, 33, 2, 771, 2, 35, 73, 331, 20, 422, 2, 198, 28, 216, 2, 1369

Offset: 0

Gus Wiseman, Apr 22 2025

nonn

			The partition (4,4,2,2,2,2,1,1,1,1,1,1,1,1) has two partitions into constant blocks with a common sum: {{4,4},{2,2,2,2},{1,1,1,1,1,1,1,1}} and {{4},{4},{2,2},{2,2},{1,1,1,1},{1,1,1,1}}, so is counted under a(24).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (2211)               (2222)
                                     (3111)               (22211)
                                     (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)

Twice-partitions of this type (constant with common) are counted by A279789.
Multiset partitions of this type are ranked by A383309.
The complement is counted by A381993, ranks A381871.
For sets we have the complement of A381994, see A381719, A382080.
Normal multiset partitions of this type are counted by A382203, sets A381718.
For distinct instead of equal block-sums we have A382427.
These partitions are ranked by A383014, nonzeros of A381995.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers, see A381715.
A323774 counts partitions into constant blocks with a common sum
Constant blocks with distinct sums: A381635, A381636, A381717.
Permutation with equal run-sums: A383096, A383098, A383100, A383110
Cf. A006171, A047966, A279784, A295935, A323774, A326534, A353864, A355743, A381716, A382076, A382204.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]>0&]],{n,0,30}]

Multiset systems of this type have MM-numbers A383309 = A326534 /\ A355743.

Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.

More terms from Jakub Buczak, May 03 2025

A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 7, 9, 11, 13, 16, 19, 22, 26, 32, 36, 42, 52, 59, 66, 79, 93, 108, 125, 141, 162, 192, 222, 248, 285, 331, 375, 430, 492, 555, 632, 719, 816, 929, 1051, 1177, 1327, 1510, 1701, 1908, 2146, 2408, 2705, 3035, 3388, 3792, 4257, 4751, 5284, 5894

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(9) = 11 rooted twice-partitions:
(7), (1111111),
(6)(), (33)(), (222)(), (111111)(), (11111)(1), (22)(2), (1111)(11),
(1111)(1)(), (111)(11)().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002865, A032305, A047966, A063834, A093637, A296134, A300383, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Total/@#&&SameQ@@Join@@#&]//Length,{n,20}]

a(n)=if(n<3, 1, sum(k=1, n-2, polcoef(prod(j=0, (n-2)\k, 1 + x^(j*k + 1) + O(x^n)), n-1))) \\ Andrew Howroyd, Aug 26 2018

Terms a(26) and beyond from Andrew Howroyd, Aug 26 2018

A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 9, 14, 16, 25, 30, 41, 52, 69, 83, 105, 129, 164, 208, 263, 315, 388, 449, 573, 694

Offset: 0

Gus Wiseman, Mar 26 2025

			The a(4) = 3 through a(8) = 14 partitions and their unique multiset partition into constant blocks with distinct sums:
  {4}     {5}       {6}         {7}        {8}
  {22}    {1}{4}    {33}        {1}{6}     {44}
  {1}{3}  {2}{3}    {1}{5}      {2}{5}     {1}{7}
          {11}{3}   {2}{4}      {3}{4}     {2}{6}
          {1}{22}   {11}{4}     {11}{5}    {3}{5}
          {2}{111}  {11}{22}    {1}{33}    {11}{6}
                    {1}{2}{3}   {3}{22}    {2}{33}
                    {1}{11}{3}  {1}{2}{4}  {11}{33}
                                {3}{1111}  {11}{222}
                                           {1}{2}{5}
                                           {1}{3}{4}
                                           {1}{3}{22}
                                           {1}{4}{111}
                                           {1}{111}{22}

For distinct blocks instead of block-sums we have A000726, ranks A004709.
Twice-partitions of this type (constant with distinct) are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
For no choices we have A381717, ranks A381636, zeros of A381635.
The Heinz numbers of these partitions are A381991, positions of 1 in A381635.
Normal multiset partitions of this type are counted by A382203.
For at least one choice we have A382427.
For strict instead of constant blocks we have A382460, ranks A381870.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A293511, A295935, A353864, A381633, A381716, A381990, A381992, A381993, A382079.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,10}]

A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 14, 19, 28, 39, 50, 70, 91, 120, 161, 203, 260, 338, 426, 556, 695, 863, 1082, 1360, 1685

Offset: 0

Gus Wiseman, Mar 26 2025

Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.

			The partition (3,2,2,2,1) can be partitioned as {{1},{2},{3},{2,2}} or {{1},{3},{2,2,2}}, so is counted under a(10).
The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (1111)  (221)    (51)      (61)
                            (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
The complement is counted by A381717, ranks A381636, zeros of A381635.
For strict instead of constant blocks we have A381992, ranks A382075.
For a unique choice we have A382301, ranks A381991.
Normal multiset partitions of this type are counted by A382203, sets A381718.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A295935, A300385, A353864, A381633, A381716, A381990, A381993, A382079, A382876.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]

A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 6, 2, 10, 3, 6, 2, 24, 2, 6, 4, 17, 2, 36, 2, 18, 4, 6, 2, 86, 3, 6, 10, 18, 2, 44, 2, 50, 4, 6, 4, 159, 2, 6, 4, 62, 2, 44, 2, 18, 30, 6, 2, 486, 3, 12, 4, 18, 2, 140, 4, 62, 4, 6, 2, 932, 2, 6, 30, 157, 4, 44, 2, 18, 4, 20, 2, 1500, 2, 6

Offset: 0

Gus Wiseman, Apr 03 2025

nonn

These are strict twice-partitions of weight n and type PRR.

			The a(1) = 1 through a(8) = 10 twice-partitions:
  (1)  (2)   (3)    (4)      (5)      (6)       (7)        (8)
       (11)  (111)  (22)     (11111)  (33)      (1111111)  (44)
                    (1111)            (222)                (2222)
                    (11)(2)           (111111)             (22)(4)
                    (2)(11)           (111)(3)             (4)(22)
                                      (3)(111)             (1111)(4)
                                                           (4)(1111)
                                                           (11111111)
                                                           (1111)(22)
                                                           (22)(1111)

Gus Wiseman, Sequences enumerating triangles of integer partitions

For distinct instead of equal block-sums we have A279786.
This is the strict case of A279789.
The orderless version is A304442, see A353833, A381995, A381871.
Multiset partitions of this type are ranked by A326534 /\ A355743 /\ A005117.
Partitions with no partition of this type are counted by A382076, strict case of A381993.
Normal multiset partitions of this type are counted by the strict case of A382204.
A006171 counts multiset partitions into constant blocks of integer partitions of n.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000005, A000040, A000688, A018818, A047966, A063834, A260685, A279784, A356065, A381453, A381455.

Table[If[n==0,1,Sum[Binomial[Length[Divisors[n/d]],d]*d!,{d,Divisors[n]}]],{n,0,100}]

a(n) = Sum_{d|n} binomial(A000005(n/d),d) * d!

A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 17, 32, 54, 100, 166, 289, 494, 840, 1393, 2400, 3931, 6498, 10861, 17728, 28863, 47557, 77042, 123881, 201172, 322459, 517032, 827993, 1316064, 2084632, 3328204, 5236828, 8247676, 13005652, 20417628, 31934709, 49970815, 77789059, 121144373

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 17 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(4)(), (31)(), (22)(), (211)(), (1111)(), (3)(1), (21)(1), (111)(1),
(2)(1)(), (11)(1)().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002865, A032305, A063834, A093637, A271619, A281113, A296118, A300383, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=x*Product[1+PartitionsP[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(prod(k=1, n-1, 1 + numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} (1 + A000041(n-1) x^n).

A301763 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 5, 8, 13, 14, 5, 32, 7, 20, 64, 26, 6, 92, 7, 126, 199, 22, 5, 352, 252, 41, 581, 394, 7, 1832, 9, 292, 2119, 31, 3216, 4946, 10, 40, 8413, 7708, 9, 20656, 9, 2324, 53546, 24, 5, 70040, 16395, 59361, 131204, 9503, 7, 266780, 178180, 82086

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 8 rooted twice-partitions: (5), (11111), (2)(2), (2)(11), (11)(2), (11)(11), (1)(1)(1), ()()()()()().
The a(15) = 20 rooted twice-partitions:
()()()()()()()()()()()()()(),
(1)(1)(1)(1)(1)(1)(1), (111111)(111111), (1111111111111),
(111111)(222), (222)(111111), (222)(222),
(111111)(33), (222)(33), (33)(111111), (33)(222), (33)(33),
(111111)(6), (222)(6), (33)(6), (6)(111111), (6)(222), (6)(33), (6)(6),
(13).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A002865, A047968, A063834, A093637, A127524, A295924, A300383, A301422, A301462, A301467, A301480, A301706.

Table[If[n===1,1,Sum[If[d===n-1,1,DivisorSigma[0,(n-1)/d-1]]^d,{d,Divisors[n-1]}]],{n,50}]

a(n)=if(n==1, 1, sumdiv(n-1, d, if(d==n-1, 1, numdiv((n-1)/d-1)^d))) \\ Andrew Howroyd, Aug 26 2018

A318915 Number of joining pairs of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 33, 41, 77, 105, 173, 215, 381, 449, 699, 911, 1335, 1611, 2433, 2867, 4179, 5113, 6903, 8251, 11769, 13661, 18177, 22011, 28997, 33711, 45251

Offset: 0

Gus Wiseman, Sep 05 2018

Two integer partitions are a joining pair if they have no common cover (coarser partition) other than the maximum. For example, (221) and (311) are not a joining pair as they are both covered by (32) or (41), while (222) and (33) are a joining pair.
All terms are odd.
The same as the number of pairs of integer partitions of n without common subsums. - Mamuka Jibladze, Jun 16 2024

			The sequence of joining pairs of integer partitions begins:
  ()()   (1)(1)   (2)(2)    (3)(3)     (4)(4)      (5)(5)
                  (2)(11)   (3)(21)    (4)(31)     (5)(41)
                  (11)(2)   (3)(111)   (4)(22)     (5)(32)
                            (21)(3)    (4)(211)    (5)(311)
                            (111)(3)   (4)(1111)   (5)(221)
                                       (31)(4)     (5)(2111)
                                       (31)(22)    (5)(11111)
                                       (22)(4)     (41)(5)
                                       (22)(31)    (41)(32)
                                       (211)(4)    (32)(5)
                                       (1111)(4)   (32)(41)
                                                   (311)(5)
                                                   (221)(5)
                                                   (2111)(5)
                                                   (11111)(5)

P. Erdős, J. Nicolas and A. Sárközy, On the number of pairs of partitions of n without common subsums, Colloquium Mathematicae, 63 (1992), 61-83.

Cf. A000041, A059849, A060639, A181939, A265947, A299925, A300383, A317141, A317143.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
ptncaps[y_]:=Union[Map[Sort[Total/@#,Greater]&,mps[y],{1}]];
Table[Select[Tuples[IntegerPartitions[n],2],Intersection@@ptncaps/@#=={{n}}&]//Length,{n,6}]

a(n) >= 2 * A000041(n) - 1. - Alois P. Heinz, Sep 06 2018

a(13)-a(30) from Alois P. Heinz, Sep 05 2018

cp's OEIS Frontend

A302094 Number of relatively prime or monic twice-partitions of n.

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A316091 Heinz numbers of integer partitions of prime numbers.

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A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

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A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

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A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

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A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

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A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.

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A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

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